Volume of a solid between two surfaces
If \(f\left( {x,y} \right) \ge g\left( {x,y} \right)\) over a region \(R\), then the volume of the solid between the surfaces \({z_1}\left( {x,y} \right)\) and \({z_2}\left( {x,y} \right)\) over \(R\) is given by
\(V = {\large\iint\limits_R\normalsize} {\big[ {f\left( {x,y} \right) }}\) \(-\;{{ g\left( {x,y} \right)} \big]dA} \)

Area and volume in polar coordinates
Let a region \(S\) be given in polar coordinates in the \(xy\)-plane and bounded by the lines \(\theta = \alpha,\) \(\theta = \beta,\) \(r = h\left( \theta \right),\) \(r = g\left( \theta \right).\) Let also a function \(f\left( {r,\theta} \right)\) be given in the region \(S\). Then the area of the region \(S\) and volume of the solid bounded by the surface \(f\left( {r,\theta} \right)\) are determined by the formulas
\(A = {\large\iint\limits_S\normalsize} {dA} =\) \({\large\int\limits_\alpha ^\beta\normalsize} {{\large\int\limits_{h\left( \theta \right)}^{g\left( \theta \right)}\normalsize} {rdrd\theta } } ,\) \(V = {\large\iint\limits_S\normalsize} {f\left( {r,\theta } \right)rdrd\theta } \)

Mass of a lamina
\(m = {\large\iint\limits_R\normalsize} {\rho \left( {x,y} \right)dA},\)
where the lamina occupies the region \(R\) and its density at a point \({\left( {x,y} \right)}\) is \({\rho \left( {x,y} \right)}.\)

Static moments of a lamina
The static moment of a lamina about the \(x\)-axis is given by the formula
\({M_x} = {\large\iint\limits_R\normalsize} {y\rho \left( {x,y} \right)dA} \)
Similarly, the static moment of a lamina about the \(y\)-axis is expressed in the form
\({M_y} = {\large\iint\limits_R\normalsize} {x\rho \left( {x,y} \right)dA} \)

Moments of inertia of a lamina
The moment of inertia about the \(x\)-axis is given by
\({I_x} = {\large\iint\limits_R\normalsize} {{y^2}\rho \left( {x,y} \right)dA} \)
The moment of inertia about the \(y\)-axis is determined by the formula
\({I_y} = {\large\iint\limits_R\normalsize} {{x^2}\rho \left( {x,y} \right)dA} \)
The polar moment of inertia is equal to
\({I_0} = {\large\iint\limits_R\normalsize} {\left({x^2} + {y^2}\right) }\) \({\rho \left( {x,y} \right)dA} \)

Charge of a plate
\(Q = {\large\iint\limits_R\normalsize} {\sigma \left( {x,y} \right)dA}, \)
where the electrical charge is distributed over the region \(R\) and its density at a point \({\left( {x,y} \right)}\) is \({\sigma \left( {x,y} \right)}.\)